Construction of Hadamard states by pseudo-differential calculus
Quantum field theory on curved spacetimes is a generalization of the theory of quantum fields on (flat) Minkowski space \(\mathbb {R}^4\) to a fixed curved background described as a globally hyperbolic Lorentzian manifold. One of the essential features of this more general setting is the absence of full translational invariance. This fact makes the selection of physical states over the observable algebra a challenging problem. Hadamard states are by now a widely accepted class of possible physical states which allow, e.g., the renormalization of the stress-energy tensor in a natural way. Since the work by \textit{M. J. Radzikowski} [Commun. Math. Phys. 179, No. 3, 529--553 (1996; Zbl 0858.53055)], techniques from micro-local analysis allowed a new characterization of Hadamard states in terms of conditions on the wave-front set of the corresponding two-point function. Using pseudo-differential operators Junker gave a procedure to construct Hadamard states for Klein-Gordon fields on spacetimes with a compact Cauchy surface [\textit{W. Junker}, Rev. Math. Phys. 8, No. 8, 1091--1159 (1996; Zbl 0869.53053)]. The paper under review presents a new construction of quasi-free Hadamard states for Klein-Gordon field on a class of spacetimes whose Lorentzian metric is well-behaved at space-like infinity. In particular, the authors construct all pure Hadamard states whose two-point function is a matrix of pseudo-differential operators and study some of their invariance properties. The key technical ingredient of this approach is the construction of a parametrix for the corresponding Cauchy problem.
quantum field theory on curved spacetime
scientific article; zbMATH DE number 6249795
scientific article